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Pearson Algebra 2 Common Core, 2011 View details

5. Systems With Three Variables

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Exercise 24 Page 172

When using the Substitution Method to solve a system of equations, it is necessary to isolate a variable.

(5,2,2)

Practice makes perfect

The given system consists of equations of planes. When using the Substitution Method to solve a system of equations, it is necessary to isolate a variable. In the third equation, z is already isolated, so we will start by substituting its value into the second equation.

13=3x-y 4y-3x+2z=-3 z=2x-4y

Substitute

(II): z= 2x-4y

13=3x-y 4y-3x+2( 2x-4y)=-3 z=2x-4y

▼

(II): Solve for x

Distr

(II): Distribute 2

13=3x-y 4y-3x+4x-8y=-3 z=2x-4y

AddSubTerms

(II): Add and subtract terms

13=3x-y x-4y=-3 z=2x-4y

AddEqn

(II):LHS+4y=RHS+4y

13=3x-y x=4y-3 z=2x-4y

This time, the x-variable was isolated in the second equation. We can now substitute its equivalent expression into the first equation.

13=3x-y x=4y-3 z=2x-4y

Substitute

(I): x= 4y-3

13=3( 4y-3)-y x=4y-3 z=2x-4y

▼

(I): Solve for y

Distr

(I): Distribute 3

13=12y-9-y x=4y-3 z=2x-4y

SubTerm

(I): Subtract term

13=11y-9 x=4y-3 z=2x-4y

AddEqn

(I): LHS+9=RHS+9

22=11y x=4y-3 z=2x-4y

DivEqn

(I): .LHS /11.=.RHS /11.

2=y x=4y-3 z=2x-4y

RearrangeEqn

(I): Rearrange equation

y=2 x=4y-3 z=2x-4y

The value of y is 2. Substituting 2 for y into the second equation, we can find the value of x.

y=2 x=4y-3 z=2x-4y

Substitute

(II): y= 2

y=2 x=4( 2)-3 z=2x-4y

Multiply

(II):Multiply

y=2 x=8-3 z=2x-4y

SubTerm

(II): Subtract term

y=2 x=5 z=2x-4y

Now that we know the values of x and y, we are able to find the value of z.

y=2 x=5 z=2x-4y

SubstituteII

(III): x= 5, y= 2

y=2 x=5 z=2( 5)-4( 2)

▼

(III): Evaluate right-hand side

Multiply

(III): Multiply

y=2 x=5 z=10-8

SubTerm

(III): Subtract term

y=2 x=5 z=2

The solution to the system is the point ( 5, 2, 2). This is the singular point at which all three planes intersect. Let's check our solution by substituting the values into the system.

13=3x-y 4y-3x+2z=-3 z=2x-4y

(I), (II), (III): Substitute values

13? =3( 5)- 2 4( 2)-3( 5)+2( 2)? =-3 2? =2( 5)-4( 2)

(I), (II), (III): Multiply

13? =15-2 8-15+4? =-3 2? =10-8

(I), (II), (III): Add and subtract terms

13=13 ✓ -3=-3 ✓ 2=2 ✓

Since the substitution of our answers into the given equations resulted in three identities, we know that our solution is correct.

Subchapter links

Lesson Check p.171

Practice and Problem-Solving Exercises p.171-173

Standardized Test Prep p.173

Mixed Review p.173

Exercise 24 Page 172

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